Stationary states of chains of oscillators

For the case of homogeneous chains of harmonic oscillators, analytic or semi-analytic results for the quantum-mechanical eigenenergies of cyclic or linear systems, respectively, are available. Hence, we use them here as a testing ground for the conventional numerical schemes implemented in WavePacket versus the tensor train based techniques implemented in WaveTrain.

Cyclic chains

For cyclic, homogeneous chains of N sites held together by harmonic oscillators, analytic solutions of the corresponding time-independent Schrödinger equation are well-known, see e.g. the corresponding Wikipedia article on the quantum harmonic oscillator and/or our publication on coupled excitons and phonons. For q being the wavenumber of an acoustic phonon in a one-dimensional lattice, the energy levels are obtained as

where the n_q≥0 are integer phonon quantum numbers. The corresponding (normal mode) frequencies Ω_q are given by

with lattice constant a. The (dimensionless) wavenumbers q_ja=2πj/N are restricted to discrete values −N/2+1 ≤ j ≤ N/2 (for N even) or −(N-1)/2 ≤ j ≤ (N-1)/2 (for N odd).

For the example of a cyclic homogeneous hexamer (N=6), the possible (dimensionless) wavenumbers qa are ± π/3, ± 2π/3, and ± π. And for ω=√2ν the zero point energy is found to be E₀/ν= 5/2 + √2 + ½ √5 = 5.0322475511 ...

The conventional grid-based WavePacket software package using a Gauss-Hermite grid with 8 points for each degree of freedom and imaginary time propagation for 4000 (atomic) units of time reproduces the analytical result very well within 40 seconds of CPU time: E₀/ν = 5.0322472449

• Animated wavepacket
• Input data file
• Logfile output

The tensor-train based WaveTrain software package using standard parameters (ranks=20, repeats=20) and second quantization with 8 basis functions for each degree of freedom reproduces the analytic result very precisely within 17 seconds of CPU time: E₀/ν = 5.0322475633

• Graphical output
• Input script
• Output file

Linear chains

For linear, homogeneous chains of harmonic oscillators, fully analytic solutions of the corresponding time-independent Schrödinger equation are not available. Instead, the vibrational frequencies are obtained from a conventional normal mode analysis as the square roots of the eigenvalues of the Hessian matrix of our vibrational model Hamiltonian. Because this matrix is tridiagonal and symmetric, it can be easily diagonalized with high accuracy.

For the example of a linear homogeneous hexamer (N=6) the possible (dimensionless) wavenumbers are q_ja = πj/(N+1) with 1≤j≤N. And for ω=√2ν the semi-analytic result for the zero point energy is found to be
E₀/ν= 4.7238123089 ...

The conventional grid-based WavePacket software package using a Gauss-Hermite grid with 8 points for each degree of freedom and imaginary time propagation for 4000 (atomic) units of time reproduces the analytical result very well: E₀/ν = 4.7238121503 ...

• Animated wavepacket
• Input data file
• Logfile output

The tensor-train based WaveTrain software package using standard parameters (ranks=20, repeats=20) and second quantization with 8 basis functions for each degree of freedom reproduces the analytic result very precisely: E₀/ν = 4.7238123106 ...

• Graphical output
• Input script
• Output file

Excited states

Here we switch to the case of a cyclic trimer. The conventional grid-based WavePacket software package using a standard FFT grid with 24x24x24 points can still do a diagonalization of the (sparse) Hamiltonian matrix in less than a minute. In addition to the ground state, numerous excited states are available of which we are showing the first eight here.

• Animated sequence of eigenfunctions
• Input data file
• Logfile output

Note that also the WaveTrain software package can calculate vibrationally excited states because we introduced an approach that directly incorporates the Wielandt deflation technique into the alternating linear scheme for the solution of eigenproblems, see our publication on coupled excitons and phonons.

Scaling behavior

While the conventional, grid-based numerical schemes implemented in WavePacket were successful in finding the vibrational ground state for N=6 for the examples given above, it should be mentioned, however, that on a standard PC one cannot go beyond N=7 or N=8 due to the exponential growth of the computational effort.

This is in marked contrast to the tensor-train based techniques implemented in WaveTrain which can handle chains comprising tens of sites, see our publication on coupled excitons and phonons. By employing efficient low-rank tensor-train decompositions, the required storage is found to increase only linearly with the chain length, and the CPU time is found to increase only slightly faster than linearly, thus mitigating the curse of dimensionality.